Séminaire de Mathématique

# Modular Iwahori-Hecke algebras: A survey and some computations

## par Prof. Peter Schneider (Universität Münster)

Europe/Paris
Amphithéâtre Léon Motchane (IHES)

### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description
Avec le soutien de :


ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

The smooth representation theory of a p-adic reductive group G with characteristic zero coefficients is very closely connected to the module theory of its (pro-p) Iwahori-Hecke algebra H = H(G). In the modular case,where the coefficients have characteristic p, this connection breaks down to a large extent. In this talk I will first survey joint work with R. Ollivier in this modular case.

We determine completely the homological properties of H, and we introduce a certain torsion theory in the module category Mod(H) such that the torsion free modules embed fully faithfully into the category of smooth    G-representations. In the case of the group SL_2 we are able to explicitly compute this torsion theory. Secondly I will describe a derived picture of the whole situation in which one recovers an equivalence between the module theory of a derived version of H and the derived representation theory of G. In both approaches  the cohomology of the pro-p Iwahori subgroup of G in a certain universal module plays a crucial role.

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